NCERT’s Class 9 Mathematical Solutions incorporates solutions to all questions provided in the NCERT Class 9 textbook. Students can download a PDF of the chapter-specific solutions to these issues, from the links provided at the bottom of this page. These NCERT Class 9 solutions cover all topics included in the NCERT textbook-like Number System, Coordinate Geometry, Polynomials, Euclid’s Geometry, Quadrilaterals, Triangles, Circles, Constructions, Surface Areas and Volumes, Statistics, Probability, etc.
This chapter discusses different topics, including rational numbers and irrational numbers. Students will also be learning the extended version of the number line and how to represent numbers (integers, rational and irrational) on it. A total of 6 exercises are present in this chapter that contains the problems based on all the topics asked in the chapter. This chapter also teaches students the representation of terminating/non-terminating recurring decimals (and successive magnification method) as well as the presentation of square roots of 2, 3 and other non-rational numbers on the number line. The chapter also deals with the laws of integral powers and rational exponents with positive real bases in Number System.
Review of representation of natural numbers, integers, rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.
Important Formulas –
Operations on Real Numbers\\\sqrt{ab}=\sqrt{a}\sqrt{b} \\ \\\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}ab=abba=ba
(√a + √b) (√a – √b) = a – b
(a + √b) (a – √b) = a2 – b
(√a + √b) (√c + √d) = √ac + √ad + √bc + √bd
(√a + √b)2 = a + 2√ab + b
Laws of Exponents for Real Numbers
am . an = am + n
(am)n = amn
am/an = am – n, m > n
ambm = (ab)m
This chapter discusses a particular type of algebraic expression called polynomial and terminology related to it. Polynomial is an expression that consists of variables and coefficients, involving the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The chapter also deals with Remainder Theorem and the Factor Theorem with the uses of these theorems in the factorisation of polynomials. Students will be taught several examples as well as the definition of different terms like polynomial, degrees, coefficient, zeros and terms of a polynomial. A total of 5 exercises are present in this chapter which includes problems related to all the topics mentioned in the chapter.
Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem.
Recall of algebraic expressions and identities. Verification of identities
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
(x ± y)3 = x3 ± y3 ± 3xy (x ± y)
x3 ± y3 = (x ± y) (x2 ± xy + y2)
and their use in factorization of polynomials.
Important Formulas –
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2
x2 – y2 = (x + y) (x – y)
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
(x + y)3 = x3 + y3 + 3xy(x + y)
(x – y)3 = x3 – y3 – 3xy(x – y)
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
Dividend = (Divisor × Quotient) + Remainder
Remainder theorem
Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).
Factor theorem
If p(x) is a polynomial of degree n > 1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x).
The chapter Coordinate Geometry includes the concepts of the cartesian plane, coordinates of a point in xy – plane, terms, notations associated with the coordinate plane, including the x-axis, y-axis, x- coordinate, y-coordinate, origin, quadrants and more. Students, in this chapter, will also be studying the concepts of Abscissa and ordinates of a point as well as plotting and naming a point in xy – plane. There are 3 exercises in this chapter that contain questions revolving around the topics mentioned in the chapter, helping the students get thorough with the concepts.
The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.
Important Points –
Along with recalling the knowledge of linear equations in one variable, this chapter will introduce the students to the linear equation in two variables, i.e., ax + by + c = 0. Students will also learn to plot the graph of a linear equation in two variables. There are 4 exercises in this chapter that consist of questions related to finding the solutions of a linear equation, plotting a linear equation on the graph and other topics discussed in the chapter.
Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax+by+c=0. Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line. Graph of linear equations in two variables. Examples, problems from real life with algebraic and graphical solutions being done simultaneously.
Important Points –
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